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%I #19 Sep 11 2022 09:29:56
%S 1,1,1,1,2,2,1,3,6,0,1,4,12,16,16,1,5,20,50,90,80,1,6,30,108,300,552,
%T 516,1,7,42,196,742,2100,3990,3794,1,8,56,320,1536,5888,16976,32656,
%U 31456,1,9,72,486,2826,13680,53046,154350,299628,290970,1,10,90,700,4780,27960,136380,532340,1559040,3044900,2974380
%N Triangle read by rows: T(n,k) is the number of permutations of k elements from [1..n] with longest consecutive chain size less than 3, when 1 and n are considered to be consecutive.
%C In a convex n-gon, the number of paths using k non-repeated vertices and fewer than 3 vertices (2 sides) in a row.
%F T(n,k) = n*(A340106(n-1,k-1) - S(n-2,k-2)) except for T(n,0)=1, where S(n,k) = 2*A340106(n-1,k-1) - 2*A340106(n-2,k-2) + S(n-3,k-3), S(n,k)=0 for k <= 0. [exception added by _Xiangyu Chen_, Aug 19 2022]
%e n\k 0 1 2 3 4 5 6 7 8
%e 0 1
%e 1 1 1
%e 2 1 2 2
%e 3 1 3 6 0
%e 4 1 4 12 16 16
%e 5 1 5 20 50 90 80
%e 6 1 6 30 108 300 552 516
%e 7 1 7 42 196 742 2100 3990 3794
%e 8 1 8 56 320 1536 5888 16976 32656 31456
%Y Cf. A338526, A338838, A338849.
%Y Cf. A340106, A340108.
%K nonn,tabl
%O 0,5
%A _Xiangyu Chen_, Dec 28 2020